VN-TST 1985

The sequence $(x_n)$ of real numbers is defined by $x_1=\frac{29}{10}$ and $x_{n+1}=\frac{x_n}{\sqrt{x_n^2-1}}+\sqrt{3}$ for all $n\ge 1$ . Find a real number $a$ (if exists) such that $x_{2k-1}>a>x_{2k}$ .

Let $ABC$ be a triangle with $AB = AC$ . A ray $Ax$ is constructed in space such that the three planar angles of the trihedral angle $ABCx$ at its vertex $A$ are equal. If a point $S$ moves on $Ax$ , find the locus of the incenter of triangle $SBC$ .

Does there exist a triangle $ABC$ satisfying the following two conditions: (a) ${ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C}$ (b) $S\ge a^2 - (b - c)^2$ where $S$ is the area of the triangle $ABC$ .

A convex polygon $A_1,A_2,\cdots ,A_n$ is inscribed in a circle with center $O$ and radius $R$ so that $O$ lies inside the polygon. Let the inradii of the triangles $A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n - 1}A_n$ be denoted by $r_1,r_2,\cdots ,r_{n - 2}$ . Prove that $r_1 + r_2 + ... + r_{n - 2}\leq R(n\cos \frac {\pi}{n} - n + 2)$ .

Find all real values of a for which the equation $(a - 3x^2 + \cos \frac {9\pi x}{2})\sqrt {3 - ax} = 0$ has an odd number of solutions in the interval $[ - 1,5]$

Suppose a function $f: \mathbb R\to \mathbb R$ satisfies $f(f(x)) = - x$ for all $x\in \mathbb R$ . Prove that $f$ has infinitely many points of discontinuity.